PERTURBATION METHOD FOR THE EIGENVALUE PROBLEM OF LIGHTLY DAMPED SYSTEMS

An efficient method for the determination of the eigenvalues and eigenvectors of lightly damped systems is developed by means of a perturbation technique. The second order matrix differential equation containing mass, stiffness and damping matrices is normally transformed into a first-order state equation to deal with the general damping matrix. However, the method described in this paper enables us to predict the complex eigenvalues and eigenvectors with relative ease without forming the state equation. The light damping implies that the eigensolution of the damped system differs slightly from the eigensolution of the undamped system, which also implies that we can express the eigensolution of the lightly damped system in terms of a power series expanded from the eigensolution of the undamped system. Once the eigenvalue problem of the undamped system is solved, the higher order terms which reflect the effect of damping can be obtained from the matrix equations, which reduce to simple algebraic equations. A numerical example illustrates the effectiveness of the new method. © 1993 Academic Press Limited.